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<p>The singular value decomposition (SVD) of a reordering of a matrix <i>A</i> can be used to determine an efficient Kronecker product (KP) sum approximation to <i>A</i>. We present the use of an approximate truncated SVD (TSVD) to find the KP approximation, and contrast using a randomized singular value decomposition algorithm (RSVD), a new enlarged Golub Kahan Bidiagonalization algorithm (EGKB) and the exact TSVD. The EGKB algorithm enlarges the Krylov subspace beyond a given rank for the desired approximation. A suitable rank is determined using an automatic stopping test. We also contrast the use of single and double precision arithmetic to find the approximate TSVDs. To illustrate the accuracy and efficiency in terms of memory and computational cost of these approximate KPs, we consider the solution of the total variation regularized image deblurring problem using the split Bregman algorithm implemented in double precision. Together with an efficient implementation for the reordering of <i>A</i> we demonstrate that the approximate KP sum can be obtained using a TSVD, and that the new EGKB algorithm contrasts favorably with the use of the RSVD. These results verify that it is feasible to use single precision when estimating a KP sum from an approximate TSVD.</p>

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Enlarged GKB and RSVD for KP approximations illustrated for mixed precision \(\ell _1\) image restoration

  • 期刊论文

  • 发表日期：

    2025年5月6日

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  • Abdulmajeed Alsubhi,
  • Rosemary A. Renaut

摘要

The singular value decomposition (SVD) of a reordering of a matrix A can be used to determine an efficient Kronecker product (KP) sum approximation to A. We present the use of an approximate truncated SVD (TSVD) to find the KP approximation, and contrast using a randomized singular value decomposition algorithm (RSVD), a new enlarged Golub Kahan Bidiagonalization algorithm (EGKB) and the exact TSVD. The EGKB algorithm enlarges the Krylov subspace beyond a given rank for the desired approximation. A suitable rank is determined using an automatic stopping test. We also contrast the use of single and double precision arithmetic to find the approximate TSVDs. To illustrate the accuracy and efficiency in terms of memory and computational cost of these approximate KPs, we consider the solution of the total variation regularized image deblurring problem using the split Bregman algorithm implemented in double precision. Together with an efficient implementation for the reordering of A we demonstrate that the approximate KP sum can be obtained using a TSVD, and that the new EGKB algorithm contrasts favorably with the use of the RSVD. These results verify that it is feasible to use single precision when estimating a KP sum from an approximate TSVD.